Recent progress in subset combinatorics of groups

Abstract

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $\beta G$ of $G$ , the combinatorial derivation.